Topology and geometry of cohomology jump loci

Abstract

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, $V_k$ and $R_k$, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of $V_k$ and $R_k$ are analytically isomorphic if the group is $1$-formal; in particular, the tangent cone to $V_k$ at $1$ equals $R_k$. These new obstructions to $1$-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at $1$ to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given